Some observations are in order. First note that, in the case of infinite structures, the appearance of b.q.o. is unavoidable. Indeed, the class \(\mathfrak {S}_{\le \omega }\)  of countable chains is \(\sum {\mathfrak {D}}\)  where \(\mathfrak {D}\)  consists of the one-element poset, hence the hypotheses of the theorem are fulfilled. The class of incomparability graphs of chains consists of graphs without edges so is totally ordered by the cardinality of their vertex-set, hence is w.q.o. The fact that the class of countable chains is w.q.o., a famous conjecture of Fraïssé, was proved by Laver (1971) [1]} by means of the theory of b.q.o. Next, the result above does not say that the class \(\mathfrak {C}\)  is b.q.o whenever it is w.q.o. We do not know if this is true. Our third observation is that the class \(\mathfrak {C}\)  is not necessarily hereditary. Since the condition on modules imposes that the three-element antichain does not belong to \(\mathfrak {C}\) , if \(\mathfrak {C}\)  is hereditary, this condition imposes that \(\mathfrak {C}\)  is made of posets coverable by two chains. In this case, due to our knowledge of these posets, we prove that \(\mathfrak {C}\)  is b.q.o. if and only if \(\mathfrak {C}\)  is w.q.o. (see the equivalence \((i) \Leftrightarrow (iii)\)  in Theorem REF ) This supports the thesis of Nash-Williams 1965 [2]} p.700, asserting that "one is inclined to conjecture that most w.q.o. sets which arise in a reasonably 'natural' manner are likely to be b.q.o."
